Group #28310044 — subgroups

Subgroups — #28310044

Order 13568.

252 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291094	C₂	1	2	yes	maximal, complemented
2	#20289737	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289666	C₂	1	2	yes	maximal, complemented
4	#20289628	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273509	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	—	2	2	no	—
7	C₄⋊#11719910	C₂²	1	2	yes	—
8	#16811518	—	2	2	no	—
9	C₄×#11719910	C₂²	1	1	yes	characteristic
10	C₄×#11719910	C₂²	1	1	yes	characteristic
11	C₄×D₄₂₄	C₄	1	1	yes	characteristic
12	#16810688	C₂²	1	2	yes	—
13	#16810350	C₄	1	2	yes	—
14	C₈×#88322	C₂²	1	1	yes	characteristic
15	C₅₃×#335	C₄	1	1	yes	characteristic
16	D₅₃⋊C₁₆	—	4	4	no	—
17	C₂×#11719910	—	2	2	no	—
18	C₂×#11719910	—	2	1	no	—
19	C₂×#11719910	D₄	1	2	yes	—
20	C₂×D₄₂₄	C₈	1	1	yes	characteristic
21	C₂×C₈×D₅₃	—	2	2	no	—
22	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
23	#13133151	C₈	1	1	yes	characteristic
24	#13133139	C₈	1	2	yes	—
25	#13133129	—	2	2	no	—
26	C₄×#88322	C₂×C₄	1	1	yes	characteristic
27	C₄×#88322	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
31	C₂×C₄×D₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	#11719921	—	2	2	no	—
37	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	C₂×D₄	1	1	yes	characteristic
40	C₂×#88322	#41	1	1	yes	characteristic
41	D₄₂₄	C₁₆	1	4	yes	complemented
42	C₈×D₅₃	—	4	2	no	—
43	C₈×D₅₃	—	4	2	no	—
44	#11719910	—	2	2	no	—
45	#11719910	—	4	1	no	—
46	#11719910	—	4	2	no	—
47	C₂²×D₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	2	no	—
52	D₂₁₂	—	2	2	no	—
53	D₂₁₂	C₂×C₁₆	1	4	yes	complemented
54	C₄×D₅₃	—	4	2	no	—
55	C₄×D₅₃	—	4	2	no	—
56	C₈×C₅₃	—	2	1	no	—
57	C₈×C₅₃	—	4	2	no	—
58	C₈×C₅₃	C₂×C₁₆	1	2	yes	complemented
59	#88322	—	2	1	no	—
60	#88322	—	4	1	no	—
61	#88322	AGL(1,ℤ/8)	1	2	yes	—
62	#12062	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
64	C₂×D₅₃	—	2	4	no	—
65	C₂×D₅₃	—	4	2	no	—
66	C₂×D₅₃	—	4	2	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	#442	1	1	yes	characteristic
69	C₄×C₅₃	#444	1	1	yes	characteristic
70	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
71	C₄×C₅₃	#504	1	1	yes	characteristic
72	Dic₅₃	—	4	2	no	—
73	D₄×C₁₆	—	53	2	no	—
74	C₈×D₈	—	53	1	no	—
75	#1367	—	53	1	no	—
76	D₄⋊C₁₆	—	53	2	no	—
77	#1312	—	53	1	no	—
78	C₂×C₅₃	#1580	1	1	yes	characteristic
79	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
80	C₂×C₅₃	#2171	1	1	yes	characteristic
81	D₅₃	—	4	4	no	—
82	C₂²×C₁₆	—	106	2	no	—
83	C₄×D₈	—	53	1	no	—
84	C₈×D₄	—	53	2	no	—
85	C₄⋊C₁₆	—	53	2	no	—
86	C₂²⋊C₁₆	—	106	2	no	—
87	C₄×C₁₆	—	53	1	no	—
88	C₄×C₁₆	—	53	1	no	—
89	#335	#6157	1	1	yes	characteristic
90	D₄⋊C₈	—	53	2	no	—
91	C₈²	—	53	1	no	—
92	C₅₃	#12062	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₈	—	53	1	no	—
94	C₂²×C₈	—	106	2	no	—
95	C₄×D₄	—	53	2	no	—
96	C₂×C₁₆	—	53	2	no	—
97	C₂×C₁₆	—	106	1	no	—
98	C₂×C₁₆	—	106	2	no	—
99	C₂×C₁₆	—	212	4	no	—
100	#107	—	53	1	no	—
101	C₄⋊C₈	C₂×#6157	1	2	yes	—
102	#102	—	53	2	no	—
103	C₂²⋊C₈	—	106	2	no	—
104	C₄×C₈	—	53	1	no	—
105	C₄×C₈	—	53	1	no	—
106	C₄×C₈	C₂×#6157	1	1	yes	characteristic
107	C₂×D₄	—	53	2	no	—
108	C₂²×C₄	—	106	2	no	—
109	D₈	—	53	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	—	212	2	no	—
115	C₂×C₈	—	212	2	no	—
116	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
117	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
118	C₄⋊C₄	—	53	2	no	—
119	C₂²⋊C₄	—	106	2	no	—
120	C₄²	C₂²×#6157	1	1	yes	characteristic
121	C₁₆	—	106	2	no	—
122	C₁₆	—	212	2	no	—
123	C₁₆	—	212	1	no	—
124	C₂³	—	106	2	no	—
125	D₄	—	53	4	no	—
126	D₄	—	106	2	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	—	212	2	no	—
129	C₂×C₄	—	212	2	no	—
130	C₂×C₄	D₄×#6157	1	1	yes	characteristic
131	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
132	C₂×C₄	#13133305	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	4	2	no	—
135	C₈	—	53	2	no	—
136	C₈	—	106	1	no	—
137	C₈	—	212	1	no	—
138	C₈	D₅₃⋊C₁₆	1	2	yes	complemented
139	C₂²	—	106	4	no	—
140	C₂²	—	212	2	no	—
141	C₂²	—	212	2	no	—
142	C₂²	#16811651	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	212	2	no	—
145	C₄	#16811664	1	1	yes	characteristic
146	C₄	#16811669	1	1	yes	characteristic
147	C₄	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
148	C₄	#16811682	1	1	yes	characteristic
149	C₂	—	212	4	no	—
150	C₂	#20290313	1	1	yes	characteristic
151	C₂	#20291094	1	1	yes	characteristic
152	C₂	#20291107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291094

C₂

yes

maximal, complemented

#20289737

C₂

yes

maximal, characteristic, complemented

#20289666

C₂

yes

maximal, complemented

#20289628

C₂

yes

maximal, characteristic, complemented

#20273509

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

—

C₄⋊#11719910

C₂²

yes

—

#16811518

—

C₄×#11719910

C₂²

yes

characteristic

C₄×#11719910

C₂²

yes

characteristic

C₄×D₄₂₄

C₄

yes

characteristic

#16810688

C₂²

yes

—

#16810350

C₄

yes

—

C₈×#88322

C₂²

yes

characteristic

C₅₃×#335

C₄

yes

characteristic

D₅₃⋊C₁₆

—

C₂×#11719910

—

C₂×#11719910

—

C₂×#11719910

D₄

yes

—

C₂×D₄₂₄

C₈

yes

characteristic

C₂×C₈×D₅₃

—

C₄×D₂₁₂

C₂×C₄

yes

—

#13133151

C₈

yes

characteristic

#13133139

C₈

yes

—

#13133129

—

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₂₁₂

C₂×C₈

yes

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₂×C₈

yes

—

C₂×#88322

—

C₂×#88322

C₂×D₄

yes

yes

D₄₂₄

C₁₆

yes

complemented

C₈×D₅₃

—

C₈×D₅₃

—

#11719910

—

#11719910

—

#11719910

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

C₂×C₁₆

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₁₆

yes

—

—

yes

—

#12062

—

maximal, Sylow

C₂²×C₅₃

C₈×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²×C₁₆

yes

characteristic

C₄×C₅₃

#504

yes

characteristic

Dic₅₃

—

D₄×C₁₆

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

—

C₂×C₅₃

#1580

yes

characteristic

C₂×C₅₃

D₄×C₁₆

yes

characteristic

C₂×C₅₃

#2171

yes

characteristic

D₅₃

—

C₂²×C₁₆

—

106

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

#6157

yes

characteristic

D₄⋊C₈

—

C₈²

—

C₅₃

#12062

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

106

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

212

—

100

#107

—

101

C₄⋊C₈

C₂×#6157

yes

—

102

#102

—

103

C₂²⋊C₈

—

106

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₂×#6157

yes

characteristic

107

C₂×D₄

—

108

C₂²×C₄

—

106

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

212

—

115

C₂×C₈

—

212

—

116

C₂×C₈

D₅₃⋊C₈

yes

characteristic

117

C₂×C₈

D₅₃⋊C₈

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

106

—

120

C₄²

C₂²×#6157

yes

characteristic

121

C₁₆

—

106

—

122

C₁₆

—

212

—

123

C₁₆

—

212

—

124

C₂³

—

106

—

125

D₄

—

126

D₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

212

—

129

C₂×C₄

—

212

—

130

C₂×C₄

D₄×#6157

yes

characteristic

131

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

132

C₂×C₄

#13133305

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

106

—

137

C₈

—

212

—

138

C₈

D₅₃⋊C₁₆

yes

complemented

139

C₂²

—

106

—

140

C₂²

—

212

—

141

C₂²

—

212

—

142

C₂²

#16811651

yes

characteristic

143

C₄

—

144

C₄

—

212

—

145

C₄

#16811664

yes

characteristic

146

C₄

#16811669

yes

characteristic

147

C₄

C₂×D₅₃⋊C₁₆

yes

characteristic

148

C₄

#16811682

yes

characteristic

149

C₂

—

212

—

150

C₂

#20290313

yes

characteristic

151

C₂

#20291094

yes

characteristic

152

C₂

#20291107

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×D₄₂₄	C₈
cosocle	C₄×#88322	C₂³
hypercenter	#335	#6157
hypocenter	C₅₃	#12062
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#6157

commutator

C₄×C₅₃

C₂²×C₁₆

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#335

C₄

socle

C₂²×C₅₃

C₈×D₄

prime radical

C₂×D₄₂₄

C₈

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #20291094	C₂	semidirect
2. #20289737	C₂	semidirect
3. #20289666	C₂	semidirect
4. #20289628	C₂	semidirect
41. D₄₂₄	C₁₆	semidirect
53. D₂₁₂	C₂×C₁₆	semidirect
58. C₈×C₅₃	C₂×C₁₆	semidirect
92. C₅₃	#12062	semidirect
138. C₈	D₅₃⋊C₁₆	semidirect

H ≅ G/N

type

1. #20291094

C₂

semidirect

2. #20289737

C₂

semidirect

3. #20289666

C₂

semidirect

4. #20289628

C₂

semidirect

41. D₄₂₄

C₁₆

semidirect

53. D₂₁₂

C₂×C₁₆

semidirect

58. C₈×C₅₃

C₂×C₁₆

semidirect

92. C₅₃

#12062

semidirect

138. C₈

D₅₃⋊C₁₆

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20291094	2
#20289737	2
#20289666	2
#20289628	2
#20273509	2
#12062	53

subgroup

index

Subgroups — #28310044

Conjugacy classes of subgroups

Complements

Maximal subgroups